On binding energy of trions in bulk materials

We study the negatively $T^{?}$ and positively $T^{+}$ charged trions in bulk materials in the effective mass approximation within the framework of a potential model. The binding energies of trions in various semiconductors are calculated by employing Faddeev equation in configuration space. Results of calculations of the binding energies for $T^{?}$ are consistent with previous computational studies and are in reasonable agreement with experimental measurements, while the $T^{+}$ is unbound for all considered cases. The mechanism of formation of the binding energy of trions is analyzed by comparing contributions of a mass-polarization term related to kinetic energy operators and a term related to the Coulomb repulsion of identical particles.     

Regularities of many-body systems interacting by a two-body random ensemble

The ground states of all even–even nuclei have angular momentum, I, equal to zero, $I=0$, and positive parity, $\pi =+$. This feature was believed to be a consequence of the attractive short-range interaction between nucleons. However, in the presence of two-body random interactions, the predominance of $I^{\pi }=0^{+}$ ground states (0 g.s.) was found to be robust both for bosons and for an even number of fermions. For simple systems, such as d bosons, sp bosons, sd bosons, and a few fermions in single-$j$ shells for small $j$, there are a few approaches to predict and/or explain spin I ground state (I g.s.) probabilities. An empirical approach to predict I g.s. probabilities is available for general cases, such as fermions in a single-$j$ ($j>\frac{7}{2}$) or many-$j$ shells and various boson systems, but a more fundamental understanding of the robustness of 0 g.s. dominance is still out of reach. Further interesting results are also reviewed concerning other robust phenomena of many-body systems in the presence of random two-body interactions, such as the odd–even staggering of binding energies, generic collectivity, the behavior of average energies, correlations, and regularities of many-body systems interacting by a displaced two-body random ensemble.     

Orbital angular momentum of channeling radiation from relativistic electrons in thin Si crystal

We propose to use channeling radiation (CR) from relativistic electrons as a source of high energy twisted photons in the MeV range. We calculate numerically the orbital angular momentum (OAM) of radiation produced by electrons with the energies $155÷2500$ MeV for the axial and planar channeling in the thin Si crystal. We obtain that the average OAM of CR in this case is approximately $1÷6?$ per photon with the photon energies about $1÷2$ MeV.     

The GL(1|1) WZW-model: From supergeometry to logarithmic CFT

We present a complete solution of the WZW model on the supergroup $\mathrm{GL}(1|1)$. Our analysis begins with a careful study of its minisuperspace limit (“harmonic analysis on the supergroup”). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the $\mathrm{gl}(1|1)$ current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups.     

Shannon and Fisher entropies for a hydrogen atom under soft spherical confinement

We calculate Shannon and Fisher entropies in the position and momentum space, and some complexity measures for a variationally described hydrogen atom confined in soft and hard spherical boxes of varying dimension $r_c$ and selected values of strength $U_0$. We include calculations for a free particle trapped in impenetrable boxes. It is found that the Shannon entropy $S_r$ becomes negative for small cavity radii and large values of $U_0$, due to the highly localized nature of the particle. For soft confinement and small cavity dimensions, the entropies change very rapidly over short radial intervals.     

1-cocycles on the group of contactomorphisms on the supercircles S1|1 and S1|2 generalizing the Schwarzian derivative

We give an explicit construction of the $1$-cocycles of the group of contactomorphisms on the supercircle $S^{1|m}$ for $m=1,2$, with coefficients in the space of differential operators acting on $S^{1|m}$-tensor densities. We show that they satisfy properties similar to those of the super-Schwarzian derivative.     

A new construction of rational electromagnetic knots

We set up a correspondence between solutions of the Yang–Mills equations on $\mathbb{R}\times S^3$ and in Minkowski spacetime via de Sitter space. Some known Abelian and non-Abelian exact solutions are rederived. For the Maxwell case we present a straightforward algorithm to generate an infinite number of explicit solutions, with fields and potentials in Minkowski coordinates given by rational functions of increasing complexity. We illustrate our method with a nontrivial example.     

Monopoles on the Bryant–Salamon G2-manifolds

$G_2$-monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant–Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the ${\mathbb{S}}^4$ and ${\mathbb{C}\mathbb{P}}^2$. These are the first nontrivial examples of $G_2$-monopoles.Associated with each monopole there is a parameter $m\in {\mathbb{R}}^{+}$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m\in {\mathbb{R}}^{+}$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on ${\Lambda }_{-}^2\left({\mathbb{S}}^4\right)$ and on ${\Lambda }_{-}^2\left({\mathbb{C}\mathbb{P}}^2\right)$.The third Bryant–Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.